AMC 8 Daily Practice Round 11

Complete problem set with solutions and individual problem pages

Problem 22 Medium

What is the number of digits in the product 25^{16} \times 2^{38}?

  • A.

    30

  • B.

    32

  • C.

    34

  • D.

    36

  • E.

    38

Answer:C

The product 25^{16} \times 2^{38} can be simplified as follows 25^{16} \times 2^{38} = 25^{16} \times 4^{19} = (25^{16} \times 4^{16}) \times 4^3.

Since 25 \times 4 = 100, we have 25^{16} \times 4^{16} = (25 \times 4)^{16} = 100^{16}.

Thus, the expression becomes 100^{16} \times 64 = 64 \times 10^{32}.

This represents a number with 2 + 32 = 34 digits (the digits 64 contribute 2 digits, and 10^{32} adds 32 zeros). Therefore, the product 25^{16} \times 2^{38} is a 34-digit number.

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Problem 20 AMC 8 Daily Practice Round 11
Problem 21 AMC 8 Daily Practice Round 11
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Problem 24 AMC 8 Daily Practice Round 11
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